Top

Journal of Elasticity

Published in:

27-01-2020

Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model

Authors: S. Baranova, S. G. Mogilevskaya, V. Mantič, S. Jiménez-Alfaro

Published in: Journal of Elasticity | Issue 2/2020

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.

next article New Classes of Traveling Waves in a Planar Kirchhoff Beam with Nonlinear Bending Stiffness

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Other transcriptions of the author name, e.g., Vencel, Ventsel or Wentzell, can also be found in the literature.

Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975) MathSciNetMATH

Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978) MATH

Miller, R.E.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11(3), 139–147 (2000) ADS

Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82(4), 535 (2003) ADS

Duan, H.L., Wang, J., Huang, Z.P., Luo, Z.Y.: Stress concentration tensors of inhomogeneities with interface effects. Mech. Mater. 37(7), 723–736 (2005)

Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L.: Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J. Mech. Phys. Solids 53(7), 1574–1596 (2005) ADSMathSciNetMATH

He, L.H., Li, Z.R.: Impact of surface stress on stress concentration. Int. J. Solids Struct. 43(20), 6208–6219 (2006) MATH

Lim, C.W., Li, Z.R., He, L.H.: Size dependent, non-uniform elastic field inside a nano-scale spherical inclusion due to interface stress. Int. J. Solids Struct. 43(17), 5055–5065 (2006) MATH

Mi, C., Kouris, D.: Nanoparticles under the influence of surface/interface elasticity. Mech. Mater. Struct. 1(4), 763–791 (2006)

10.

Mogilevskaya, S.G., Crouch, S.L., Stolarski, H.K.: Multiple interacting circular nano-inhomogeneities with surface/interface effects. J. Mech. Phys. Solids 56(6), 2298–2327 (2008) ADSMathSciNetMATH

11.

Ru, C.: Simple geometrical explanation of Gurtin-Murdoch model of surface elasticity with clarification of its related versions. Sci. China, Phys. Mech. Astron. 53(3), 536–544 (2010) ADSMathSciNet

12.

Kushch, V.I., Mogilevskaya, S.G., Stolarski, H.K., Crouch, S.L.: Elastic interaction of spherical nanoinhomogeneities with Gurtin–Murdoch type interfaces. J. Mech. Phys. Solids 59(9), 1702–1716 (2011) ADSMathSciNetMATH

13.

Kushch, V.I., Mogilevskaya, S.G., Stolarski, H.K., Crouch, S.L.: Elastic fields and effective moduli of particulate nanocomposites with the Gurtin-Murdoch model of interfaces. Int. J. Solids Struct. 50(7–8), 1141–1153 (2013)

14.

Kushch, V.I.: Stress field and effective elastic moduli of periodic spheroidal particle composite with Gurtin-Murdoch interface. Int. J. Eng. Sci. 132, 79–96 (2018) MathSciNetMATH

15.

Xu, Z., Zheng, Q.: Micro-and nano-mechanics in China: a brief review of recent progress and perspectives. Sci. China, Phys. Mech. Astron. 61(7), 1–14 (2018) ADS

16.

Zhu, Y., Wei, Y., Guo, X.: Gurtin-Murdoch surface elasticity theory revisit: an orbital-free density functional theory perspective. J. Mech. Phys. Solids 109, 178–197 (2017) ADSMathSciNet

17.

Luo, J., Wang, X.: On the anti-plane shear of an elliptic nano inhomogeneity. Eur. J. Mech. A, Solids 28(5), 926–934 (2009) ADSMATH

18.

Kushch, V.I., Chernobai, V.S., Mishuris, G.S.: Longitudinal shear of a composite with elliptic nanofibers: local stresses and effective stiffness. Int. J. Eng. Sci. 84, 79–94 (2014)

19.

Dai, M., Schiavone, P., Gao, C.: Prediction of the stress field and effective shear modulus of composites containing periodic inclusions incorporating interface effects in anti-plane shear. J. Elast. 125(2), 217–230 (2016) MathSciNetMATH

20.

Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453, 853–877 (1997) MathSciNetMATH

21.

Eremeyev, V.A., Lebedev, L.P.: Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity. Contin. Mech. Thermodyn. 28, 407–422 (2016) ADSMathSciNetMATH

22.

Hamilton, J.C., Wolfer, W.G.: Theories of surface elasticity for nanoscale objects. Surf. Sci. 603, 1284–1291 (2009) ADS

23.

Koenig, S.P., Boddeti, N.G., Dunn, M.L., Bunch, J.S.: Ultrastrong adhesion of graphene membranes. Nat. Nanotechnol. 6, 543–546 (2011) ADS

24.

Young, R.J., Liu, M., Kinloch, I.A., Li, S., Zhao, X., Vallés, C., Papageorgiou, D.G.: The mechanics of reinforcement of polymers by graphene nanoplatelets. Compos. Sci. Technol. 154, 110–116 (2018)

25.

Manikas, A.C., Pastore Carbone, M.G., Woods, C.R., Wang, Y., Souli, I., Anagnostopoulos, G., Hadjinicolaou, M., Novoselov, K.S., Galiotis, C.: Stress transfer at the nanoscale on graphene ribbons of regular geometry. Nanoscale 11, 14354–14361 (2019)

26.

Wu, C.H., Wang, M.L.: Configurational equilibrium of circular-arc cracks with surface stress. Int. J. Solids Struct. 38(24–25), 4279–4292 (2001) MATH

27.

Oh, E.-S., Walton, J.R., Slattery, J.C.: A theory of fracture based upon an extension of continuum mechanics to the nanoscale. J. Appl. Mech. 73, 792–798 (2006) ADSMATH

28.

Kim, C.I., Schiavone, P., Ru, C.Q.: The effects of surface elasticity on an elastic solid with mode-III crack: complete solution. J. Appl. Mech. 77(2), 021011 (2010) ADS

29.

Walton, J.R.: A note on fracture models incorporating surface elasticity. J. Elast. 109(1), 95–102 (2012) MathSciNetMATH

30.

Wang, X.: A mode III arc-shaped crack with surface elasticity. Z. Angew. Math. Phys. 66, 1987–2000 (2015) MathSciNetMATH

31.

Intarit, P., Senjuntichai, T., Rungamornrat, J., Rajapakse, R.K.N.D.: Penny-shaped crack in elastic medium with surface energy effects. Acta Mech. 228, 617–630 (2017) MathSciNet

32.

Zemlyanova, A.Y.: A straight mixed mode fracture with the Steigmann–Ogden boundary condition. Q. J. Mech. Appl. Math. 70(1), 65–86 (2017) MathSciNetMATH

33.

Leguillon, D., Sanchez-Palencia, E.: Computation of Singular Solutions in Elliptic Problems and Elasticity. Masson, Paris (1987) MATH

34.

Yosibash, Z.: Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation. Springer, New York (2012) MATH

35.

Mura, T.: Micromechanics of Defects in Solids. Martinus Nijhoff, Hague, Netherland (1987) MATH

36.

Melan, E.: Ein Beitrag zur Theorie geschweisster Verbindungen. Ing.-Arch. 3(2), 123–129 (1932) MATH

37.

Reissner, E.: Note on the problem of the distribution of stress in a thin stiffened elastic sheet. Proc. Natl. Acad. Sci. USA 26, 300–305 (1940) ADSMathSciNetMATH

38.

Pasternak, Y.M., Sulim, G.T.: Plane problem of elasticity for an anisotropic body with doubly periodic systems of thin inhomogeneities. Mech. Solids 49(2), 162–174 (2014) ADS

39.

Erdogan, F., Gupta, G.D.: Stresses near a flat inclusion in bonded dissimilar materials. Int. J. Solids Struct. 8(4), 533–547 (1972) MATH

40.

Caillerie, D., Nedelec, J.C.: The effect of a thin inclusion of high rigidity in an elastic body. Math. Methods Appl. Sci. 2(3), 251–270 (1980) ADSMathSciNetMATH

41.

Sulim, G.T.: Antiplane problem for a system of linear inclusions in an isotropic medium. Appl. Math. Mech. 45(2), 223–230 (1981) MathSciNetMATH

42.

Kanaun, S.K.: On singular models of a thin inclusion in a homogeneous elastic medium. J. Appl. Math. Mech. 48(1), 50–58 (1984) MATH

43.

Li, Q., Ting, T.C.T.: Line inclusions in anisotropic elastic solids. J. Appl. Mech. 56(3), 556–563 (1989) ADSMATH

44.

Vil’chevskaya, Y.N., Kanaun, S.K.: Integral equations for a thin inclusion in a homogeneous elastic medium. Appl. Math. Mech. 56(2), 235–243 (1992) MathSciNetMATH

45.

Bessoud, A., Krasucki, F., Serpilli, M.: Asymptotic analysis of shell-like inclusions with high rigidity. J. Elast. 103(2), 153–172 (2011) MathSciNetMATH

46.

Vasil’ev, K.V., Sulym, H.T.: Elastic equilibrium of a space containing a thin curved elastic inclusion in longitudinal shear. J. Math. Sci. 212(1), 83–97 (2016) MathSciNet

47.

Koiter, W.T.: On the diffusion of load from a stiffener into a sheet. Q. J. Mech. Appl. Math. 8(2), 164–178 (1955) MathSciNetMATH

48.

Muki, R., Sternberg, E.: Transfer of load from an edge-stiffener to a sheet—a reconsideration of Melan’s problem. J. Appl. Mech. 34(3), 679–686 (1967) ADS

49.

Erdogan, F., Gupta, G.D.: The problem of an elastic stiffener bonded to a half plane. J. Appl. Mech. 38, 937–941 (1971) ADSMATH

50.

Mkhitaryan, S.M., Mkrtchyan, M.S., Kanetsyan, E.G.: On a method for solving Prandtl’s integro-differential equation applied to problems of continuum mechanics using polynomial approximations. Z. Angew. Math. Mech. 97(6), 639–654 (2017) MathSciNet

51.

Delale, F., Erdogan, F.: The crack problem for a half plane stiffened by elastic cover plate. Int. J. Solids Struct. 18(5), 381–395 (1982) MATH

52.

Wang, Z.Y., Zhang, H.T., Chou, Y.T.: Stress singularity at the tip of a rigid line inhomogeneity under antiplane shear loading. J. Appl. Mech. 53(2), 459–461 (1986) ADSMATH

53.

Markenscoff, X., Ni, L., Dundurs, J.: The interface anticrack and Green’s functions for interacting anticracks and cracks/anticracks. J. Appl. Mech. 61(4), 797–802 (1994) ADSMATH

54.

Dal Corso, F., Bigoni, D., Gei, M.: The stress concentration near a rigid line inclusion in a prestressed, elastic material. Part I. Full field solution and asymptotics. J. Mech. Phys. Solids 56(3), 815–838 (2008) ADSMathSciNetMATH

55.

Simonenko, I.B.: Electrostatic problems for non uniformal media. The case of a thin dielectric with a high dielectric constant. I and II. Differ. Equ. 10, 223–229 (1975), 11, 1398–1404 (1974) MATH

56.

Benveniste, Y., Miloh, T.: Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech. Mater. 33(6), 309–323 (2001)

57.

Zemlyanova, A.Y., Mogilevskaya, S.G.: Circular inhomogeneity with Steigmann–Ogden interface: local fields, neutrality, and Maxwell’s type approximation formula. Int. J. Solids Struct. 135, 85–98 (2018)

58.

Muskhelishvili, N.: Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics, 2nd edn. Aeronautical Research Laboratories Translation, vol. 12. Dept. of Supply and Development, Melbourne (1949). Translated by J.R.M. Radok and W.G. Woolnough MATH

59.

Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., Wang, T.: Surface stress effect in mechanics of nanostructured materials. Acta Mech. Solida Sin. 24(1), 52–82 (2011)

60.

Martin, P.: End-point behaviour of solutions to hypersingular integral equations. Proc. R. Soc., Math. Phys. Sci. 432(1885), 301–320 (1991) ADSMathSciNetMATH

61.

Martin, P.: Exact solution of a simple hypersingular integral equation. J. Integral Equ. Appl. 4(2), 197–204 (1992) MathSciNetMATH

62.

Mandal, B.N., Chakrabarti, A.: Applied Singular Integral Equations. CRC Press and Science Publishers, Boca Raton (2011) MATH

63.

Linkov, A., Mogilevskaya, S.: On the theory of complex hypersingular equations. In: Atluri, S.N., Yagawa, G., Cruse, T. (eds.) Computational Mechanics ’95, pp. 2836–2840. Springer, Berlin, Heidelberg (1995)

64.

Linkov, A.M.: Boundary Integral Equations in Elasticity Theory. Kluwer Academic Publishers, Dordrecht (2002) MATH

65.

Linkov, A.M., Mogilevskaya, S.G.: Complex hypersingular BEM in plane elasticity problems. In: Sladek, V., Sladek, J. (eds.) Singular Integrals in Boundary Element Method, pp. 299–364. Computational Mechanics Publication, Southampton (1998). Chapter 9

66.

Aleksandrov, V.M., Mkhitaryan, S.M.: Contact Problems for Bodies with Thin Coatings and Interlayers. Nauka, Moscow (1983). In Russian

67.

Goldstein, G.R.: Derivation and physical interpretation of general boundary conditions. Adv. Differ. Equ. 11(4), 457 (2006) MathSciNetMATH

68.

Nicaise, S., Li, H., Mazzucato, A.: Regularity and a priori error analysis of a Ventcel problem in polyhedral domains. Math. Methods Appl. Sci. 40, 1625–1636 (2017) ADSMathSciNetMATH

69.

Bonnaillie-Noël, V., Dambrine, M., Hérau, F., Vial, G.: On generalized Ventcel’s type boundary conditions for Laplace operator in a bounded domain. SIAM J. Math. Anal. 42(2), 931–945 (2010) MathSciNetMATH

70.

Ventcel, A.D.: On boundary conditions for multi-dimensional diffusion processes. Theory Probab. Appl. 4(2), 164–177 (1959) MathSciNetMATH

71.

Costabel, M., Dauge, M.: Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems. Math. Nachr. 162(1), 209–237 (1993) MathSciNetMATH

72.

Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman Advanced Pub. Program, Boston (1985) MATH

73.

Grisvard, P.: Singularities in Boundary Value Problems. Springer, Berlin (1992) MATH

74.

Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227–313 (1967) MathSciNetMATH

75.

Golberg, M.A.: The convergence of several algorithms for solving integral equations with finite part integrals I. J. Integral Equ. 5, 329–340 (1983) MathSciNetMATH

76.

Kaya, A.C., Erdogan, F.: On the solution of integral equations with strongly singular kernels. Q. Appl. Math. XLV(1), 105–122 (1987) MathSciNetMATH

77.

Gradshteĭn, I.S., Ryzhik, I.M., Jeffrey, A., Zwillinger, D.: Table of Integrals, Series and Products, 7th edn. Academic Press, Oxford (2007)

Title: Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model
Authors: S. Baranova
S. G. Mogilevskaya
V. Mantič
S. Jiménez-Alfaro
Publication date: 27-01-2020
Publisher: Springer Netherlands
Published in: Journal of Elasticity / Issue 2/2020
Print ISSN: 0374-3535
Electronic ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-020-09764-x

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Other articles of this Issue 2/2020

Extension-Torsion-Inflation Coupling in Compressible Magnetoelastomeric Thin Tubes with Helical Magnetic Anisotropy

New Classes of Traveling Waves in a Planar Kirchhoff Beam with Nonlinear Bending Stiffness

On the Wave Propagation in the Thermoelasticity Theory with Two Temperatures

Growth and Non-Metricity in Föppl-von Kármán Shells

Generalization of Plane Stress and Plane Strain States to Elastic Plates of Finite Thickness

Spectral Properties of Neumann-Poincaré Operator and Anomalous Localized Resonance in Elasticity Beyond Quasi-Static Limit

Premium Partners